Note: Descriptions are shown in the official language in which they were submitted.
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Highly directive endfire loudspeaker array
Field of the invention
The invention relates to the field of directive endfire loudspeaker arrays.
Background of the invention
Control of the directivity of loudspeaker systems is important in applications
of sound reproduction with public address systems. The use of loudspeaker
arrays
shows great advantages to bundle the sound in specific directions. Usually, in
use, the
loudspeakers are placed on a vertical line and the directivity is mainly in a
plane
perpendicular to that line. For that purpose the loudspeakers are fed with the
same input
signal and this leads to so-called broadside beamforming. Using delays between
the
input signals to the loudspeakers, the beamforming can also be directed to
other
directions. In the extreme, the radiation direction is along the line of the
loudspeakers
and this is called endfire beamforming. Endfire beamforming is well known in
microphone array technology, but it is not often used in loudspeaker
technology,
although there are a few exceptions.
J.A. Harrell, "Constant-beamwidth one-octave bandwidth end-fire line array of
loudspeakers", J. Audio Eng. Soc., Vol. 43, No. 7/8, 1995 July/August, pp. 581-
591,
discloses such an endfire array where signals to be converted by loudspeakers
into
sound are processed with a delay and beamforming technique.
M.M. Boone and O. Ouweltjes, "Design of a loudspeaker system with a low-
frequency cardiod-like radiation pattern", J. Audio Eng. Soc., Vol. 45, No. 9,
Sept.
1997, pp. 702-707, disclose a loudspeaker system with two closely spaced
loudspeakers
arranged in an endfire arrangement. The filters used to provide the
loudspeakers with
input signals are optimized based on a gradient principle.
Summary of the invention
It is an object of the present invention to provide a loudspeaker array with
improved endfire beamforming.
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To that effect, the present invention provides a loudspeaker system as defined
in
independent claim 1.
For the case of two loudspeakers, the gradient principle as known from Boone
and Ouweltjes may be said to coincide with optimization based on super
resolution
beamforming signal processing. Therefore, the invention as claimed is
restricted to the
case where the number of loudspeakers and corresponding filters is 3 or
higher.
With a loudspeaker array thus defined a higher directivity index can be
obtained than with delay and sum beamforming.
In an embodiment, the invention provides a set of filters for an endfire array
as
defined in the claims.
Brief description of the drawings
The invention will be explained in detail with reference to some drawings that
are only intended to show embodiments of the invention and not to limit the
scope. The
scope of the invention is defined in the annexed claims and by its technical
equivalents.
The drawings show:
Figure 1 shows a general overview of a loudspeaker array with a plurality of
filters and a processor to supply the loudspeakers with an input signal;
Figures 2a and 2b show directional characteristics of arrays with different
spacings of the loudspeakers;
Figures 3a and 3b, respectively, show changes of evaluation characteristics in
dependence on number of loudspeakers for the directivity index DI and the
noise
sensitivity NS, respectively;
Figures 4a and 4b show changes of evaluation characteristics in dependence on
the value of a stability factor;
Figure 5 shows plots of a directivity index and noise sensitivity;
Figures 6a and 6b, respectively, show directivity index and noise sensitivity,
respectively, of a constant beam width array system;
Figure 7 shows a directional pattern of the system according to figures 6a and
6b;
Figures 8a and 8b, respectively, show a boundary element model for numerical
simulation for a single loudspeaker and a loudspeaker array, respectively;
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Figures 9a and 9b, respectively, show a comparison of directional
characteristics, i.e., directivity index derived by Equation (1) and the
boundary element
method, and noise sensitivity derived by Equation (5), respectively;
Figures l0a and l0b show comparisons of directivity patterns: for an actually
filter designed under simple source assumption (figure l0a), and for the same
filter
considering the directivity of the loudspeakers (figure l Ob);
Figures lla, llb, and llc show measured directional patterns of a prototype
endfire array with constant beam width: with a simple source assumption
(figure 1 l a),
using directivity of a single source obtained by a numerical model (figure
llb), and
comparisons of the directivity index (figure 11 c) for the different
assumptions.
Detailed description of embodiments
Below, results on the applicability of a loudspeaker line array are presented
where the main directivity is in the direction of that line, using so-called
endfire
beamforming, resulting in a "spotlight" of sound in a preferred direction.
Optimized
beamforming techniques are used, which were earlier developed for the
reciprocal
problem of directional microphone arrays. Effects of the design parameters of
the
loudspeaker array system are investigated and the inventor of the present
invention has
found that a stability factor can be a useful parameter to control the
directional
characteristics. A prototype constant beam width array system has been built.
Both
simulations and measurements support theoretical findings.
Directional loudspeaker systems have already been studied by many researchers
because of their useful application, e.g., a column array which addresses
sound
information in the plane of the ears of the listeners. In the case of a single
loudspeaker
unit, the directional characteristics depend on the Helmholtz number, which is
related
to the size of the radiating membrane and the wavelength. In the case of
multiple
loudspeaker units, a so-called loudspeaker array, the directional
characteristics depend
on the placement of the loudspeaker units within the array and on the
filtering of the
audio signals that are sent to the loudspeakers. A lot of work on the
behaviour of
transducer arrays has been carried out in the field of (electro-magnetic)
antennas and
also for loudspeaker and microphone systems. In recent researches, the
representative
methods to obtain highly directive beam patterns could be summarized by three
methods: delay and sum, gradient method, and optimal beamforming. Among these,
the
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optimal beamforming method is known to deliver a relatively high directivity
as
compared to other methods [l, 2]. The solution for optimal beamforming was
suggested halfway the 20th century, however, it was only considered to be of
academic
interest, because of noise problems associated with equipment [2], but also
because the
implementation of the required filters was not possible with the analogue
equipment of
that time. A constrained solution considering the noise to solve this problem
was
suggested by Gilbert and Morgan [3], and with the advent of modem digital
signal
processing equipment, this technique has been applied to many practical
situations.
One of these applications is the optimized beamforming that has been
implemented in hearing glasses [1]. These are high directivity hearing aids
mounted in
the arms of a pair of spectacles, with usually four microphones at each side.
Simulation
and measurement results on the directivity of the hearing glasses have been
presented at
the 120'h AES-convention [4].
In the invention as described below, an endfire array system is applied for
the
design and development of a highly directive loudspeaker array system. The
optimal
beamforming method is also implemented, which is usually applied in microphone
array systems. In accordance with the invention, the directivity index and the
noise
sensitivity (the inverse of the array gain) which are the most important
design
parameters of the optimal beamformer are set to an optimal value in accordance
with a
predetermined optimization criterion.
Basic theory
Evaluation of the array system
Figure 1 shows a general geometry of a loudspeaker array. The array comprises
a
plurality of loudspeakers Zõ (n = 1, 2, 3, ..., N), a plurality of filters Fõ
(n = 1, 2, 3, ...,
N), and a processor P. Each loudspeaker Zõ is connected to an associated
filter F,,. All
filters Fõ are connected to processor P. It is observed that Figure 1 only
gives a
schematic view: the circuit may be implement in many different ways. The
filters Fõ
may, for instance, be part of the processor P when the latter is implemented
as a
computer arrangement. Then, the filters Fõ are software modules in such a
computer.
However, other implementations, both digital and analogue, can be conceived.
The processor P may include a plurality of memory components, including a hard
disk, Read Only Memory (ROM), Electrically Erasable Programmable Read Only
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Memory, and Random Access Memory (RAM). Not all of these memory types need
necessarily be provided. Moreover, these memory components need not be located
physically close to the processor P but may be located remote from the
processor P.
The processor 1 may be connected to a communication network, for instance, the
5 Public Switched Telephone Network (PSTN), a Local Area Network (LAN), a Wide
Area Network (WAN). The processor P may be arranged to communicate with other
communication arrangements through such a network.
The processor P may be implemented as stand alone system, or as a plurality of
parallel operating processors each arranged to carry out subtasks of a larger
computer
program, or as one or more main processors with several sub-processors. Parts
of the
functionality of the invention may even be carried out by remote processors
communicating with processor P through the network.
In order to compare the performance of array systems, many evaluation
parameters have been suggested. The directivity factor is one of the most
important
evaluation parameters for array systems. For loudspeaker systems, the
directivity factor
is defined by the ratio of the acoustic intensity in some far field point in a
preferred
direction and the intensity obtained in the same point with a monopole source
that
radiates the same acoustic power as the array system [6]. This measure shows
how
much available acoustic power is concentrated onto the preferred direction by
the
designed system. Using the principle of acoustical reciprocity, the
directivity factor of a
loudspeaker array can be obtained by the same equation that applies for
microphone
arrays. For microphone arrays, the equation for the directivity factor is
given by [ 1]
max {FHW*WHF }
Q(~) = FHST F (1 )
For the case of a loudspeaker array, the parameters are defined as follows:
= * means the conjugate operator,
= H means the Hermitian transpose,
= F(co) is the filter array which controls the output and is connected to the
loudspeaker array:
F(w) - [F (w) Fz (~) FN (0))]T (2)
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= W(co) is the relative propagation factor from each loudspeaker Zõ to a far
field
reception point, denoted by the following vector equations of the endfire
array
system,
jcod, cos9 jcodZ cos9 jcodN cos9 T
W(cO) = T,e 0 F2e c ...TNe c (3)
Here, I'õ (n=1, 2, ...,1V) denotes the directional factor of each loudspeaker
Z,,,
and dõ = location of each loudspeaker (Zõ) relative to an origin.
= For the case of microphone arrays, S(co) is a coherence function of the
noise
field as applicable to the microphone array. If the background noise is
assumed
as uniform and isotropic, the coherence matrix S(co) is written by [1, 2]
Smn = sin [k(dm - d,z )] (4)
k(dm - dn )
where the subscripts m and n mean the index of the acoustic devices, dõ2 and
dn
are the positions of the devices relative to an origin (so, dm - dõ = distance
between two acoustic devices), and k = the wave number. Translated from
microphone to loudspeaker arrays, the coherence matrix S(co) shows the
weighting of the relevance of the radiation direction to optimize the
suppression
in certain directions. If the coherence matrix S(co) is taken uniform and
isotropic
this means that all suppression directions are taken of equal importance.
Usually, the directivity index (DI), the logarithmic value in dB of the
directivity
factor Q(co), is used. Another important evaluation parameter is the noise
sensitivity
(NS). For microphone arrays, this quantity shows the amplification ratio of
uncorrelated noise, so-called internal noise, to the signal and is given by
[1]
FH (c)) F(c~) (5)
'~(~) = FH (0)) W*(0)) WT (w)F(~)
Usually, the noise sensitivity is also expressed on a dB scale. Translating to
loudspeaker arrays the noise sensitivity transforms in a measure for the
output strength
of the array as compared to the output of a single loudspeaker unit Zõ and is
in effect
the inverse of the array gain of the array system.
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Optimal beamformer
The optimization problem of the array system is how to find a maximum
directivity index DI in combination with a minimum noise sensitivity NS. The
solution
in accordance with the invention is in applying a super resolution beamforming
signal
processing by the filters F,,. This requirement can be defined by the
following
minimization expression:
min FH((
J))ST (w )F(w), (6)
F((w)
subject to FT(c))W(c))=1.
These equations state that the output of the array system is minimized, using
a
directional weighting according to matrix S and with the constraint that the
array has
unity gain in the target (end fire) direction.
The solution of Equation (6) can be obtained by the Lagrange method and the
solution is called the minimum variance distortion less response (MVDR)
beamformer
given by the following equation for an optimal filter Fopt~mai(w), as is also
used in the
field of microphone arrays:
WH(~)S-1(~T )
F pttmal(~) WH(~)S-1(~vW(~v (7)
Unfortunately, this exact solution cannot be used in real situations due to
the high
noise sensitivity at low frequencies caused by the high condition number of
the
coherence matrix S(co) in this frequency range. To solve this mathematical
problem, in
the field of antenna arrays, Gilbert and Morgan [3] suggested adding a
stability factor,Q
to the diagonal of the coherence matrix S(co). Here, this approach as
suggested by
Gilbert and Morgan is also used. By using this method, Equation (7) can be
modified to
7, (s WH (iJ+ pI)'
FPtiõal,R - WH (S+ I)' W (g)
~'
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Optimization of design parameters
Effect of design parameters
The directional characteristics of the loudspeaker array system depend on the
array design parameters: the number of loudspeakers Z, their mutual spacing
and
distribution pattern, the directional characteristics of the single
loudspeakers Zõ and the
applied beamforming filters F,,. For the optimal beamformer, a filter shape of
the array
system is determined by Equation (8). Therefore, the parameter to be optimized
is the
stability factorfl(c)). In order to investigate the effect of each design
parameter, a
parametric study was conducted with Equations (1) and (5). Each loudspeaker Zõ
is
assumed to be a monopole and the effects of reflection and scattering are
ignored.
With uniform spacing and the same number of loudspeakers Z,,, it is observed
that the same directional characteristics apply if we normalize the
frequencies
according to the high frequency limit fh given by
fh = c / 2d , (9)
where c denotes the speed of sound and d means the spacing between two
adjacent loudspeakers Z,,.
Figures 2a and 2b, respectively, show the most important directional
characteristics, i.e., directivity index DI and noise sensitivity NS,
respectively, of arrays
which have different spacing and the same number of loudspeakers Zõ with N =
4. The
stability factor 0 is set at 0.01. The directivity index DI and noise
sensitivity NS of
these arrays coincide perfectly as a function of the normalized frequency
(i.e., relative
to fh).
The number of loudspeakers Zõ determines the maximum value of the directivity
index DI. For an endfire array system, the maximum directivity index DI is
determined
by [1]
DI,T,a~, = 201og N, (10)
where N denotes the number of loudspeakers Z,,.
Figures 3a and 3b show the results of a parametric study with,Q = 0.01.
Directivity index DI increases following the increase of N over the whole
frequency
range lower thanfh. The frequency with the maximum directivity index DI value
also
increases, but it remains belowfh. Noise sensitivity NS shows a tendency of
decreasing
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with increasing frequency and it reaches a minimum value at f= fh. These
results are in
agreement with the aforementioned theory.
Figures 4a and 4b show the change of the directional characteristics in
dependence on the stability factor,Q. Here, the number of loudspeakers Zõ is 8
and the
uniform spacing between the loudspeakers Zõis 0.15 m. With increasing,Q, the
directivity index DI and noise sensitivity NS decrease up to the frequency of
maximum
directivity index DI. At higher frequencies, directivity index DI and noise
sensitivity
NS are no longer controllable by,Q.
Optimization of the stability factor
The stability factor,Q was suggested to solve the self-noise problem of the
equipment. However, the inventor of the present invention has found that it
can also be
applied to control the directional characteristics of the array system without
changing
its configuration. The optimal value of the stability factor,Q for this
purpose cannot be
obtained by direct methods. For that reason, in the case of a microphone
array, several
iterative methods were suggested to obtain the optimal value [1]. The plot of
noise
sensitivity NS vs. directivity index DI can give useful information to
select,Q.
Consider an array system with N = 8 and d = 0.15 m which was used in the
previous section. The range of,8 is from 10-' to 10-1. Figure 5 shows the DI-
NS plot in
dependence on,Q for several frequencies. Increasing the frequency, the
variation range
of directivity index DI and noise sensitivity NS decrease with the same range
of,8. This
is related to the result of the previous section that the directional
characteristics are no
longer controllable at frequencies higher thanfh. If the target performance of
the array
system is given by a specific range of directivity index DI and noise
sensitivity NS, the
value of the stability factor can be selected on these DI-NS plots. Practical
values of
directivity index DI depend on the number of loudspeakers N. For N = 8, the
theoretical maximum is DI = 18 dB. Noise sensitivity NS will usually be kept
small,
say lower than 1 to 5, to allow sufficient acoustical output (the array gain
of the system
is inversely proportional to the noise sensitivity NS).
Example I. Constant beam width array
As an example, the inventor considered the design of a constant beamwidth
array
(CBA) system. The simplest concept to design a CBA is using the different
array sets,
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as computed for different values of the Helmholtz number kd. With this method,
however, redundant acoustic devices are required. In a specific array system,
it can be
said that the same value of directivity index DI means the same beamwidth.
Hence, the
CBA system can be designed by the selection of the frequency dependent
factor,Q(c))
5 that gives a constant directivity index DI over the whole target frequency
range.
The inventor considered an array system which has 8 loudspeakers Zõ with a
uniform spacing of 0.15 m. The directivity index DI and noise sensitivity NS
of this
system as a function of,8 are shown in Fig. 5. The target frequency range was
from 0.1
to 1 kHz and the target value of directivity index DI was 12 dB which is the
highest
10 value in Fig. 5 with noise sensitivity NS < 30 dB. To satisfy these
conditions, the,Q
values on the directivity index DI line of 12 dB were selected from Fig. 5.
The
directivity index DI and noise sensitivity NS, respectively, for the
selected,Q's are
plotted in Figures 6a and 6b, respectively. Figure 7 shows the directional
pattern of the
resulting array system. This figure shows that a constant beamwidth is
successfully
obtained within the target frequency range.
Mutual Interactions between the Loudspeakers
Directional factor of the total sound field
Up to now, the effect of reflection and scattering induced by the loudspeaker
Zõ
enclosures has been ignored (I'õ = 1, n=1, 2, ...,1V). In the case of a
microphone array
system, the size of the transducers is usually sufficiently small compared to
the
wavelength. However, for loudspeaker arrays, the size of the loudspeaker units
Zõ
should be much larger to obtain sufficient radiation power. Therefore, both
the
directivity of the single loudspeaker Zõ itself related to its own geometry
and the
system of loudspeakers Zõ owing to the scattering from the other loudspeakers
Zõ
should be considered. Usually, the scattering effect is considered as being
induced by
an incident field and the total field is described by summation of these two
sound
fields. The directional pattern of the individual loudspeakers Zõ can be found
by
summation of the direct field from the loudspeaker Zõ itself and the
scattering field
induced by the other loudspeakers Z,,. The analytical solution for the
scattered field can
be found under specific conditions [7]. However, the directional pattern of
the total
field is hard to derive theoretically, because the scattering field of each
loudspeaker Zõ
also becomes the incident field to the other loudspeakers Zõ , recursively.
For that
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reason, a numerical method or measurement is useful to obtain the directivity
of the
total sound field.
Example II: derivation of the optimal ftlters with a numerical method
As a design example, a loudspeaker array system was chosen that consists of 8
loudspeakers Zõ with 0.15 m of uniform spacing. Each loudspeaker Zõ had a
loudspeaker box and a loudspeaker diaphragm. The size of each loudspeaker box
was
0.11(W) x 0.16 (H) x 0.13 (D) m and the diameter of the loudspeaker diaphragm
was
0.075 m. The boundary element method (BEM) was applied to obtain the
directional
pattern of each loudspeaker Zõ in the given array configuration. Each
loudspeaker Zõ
was modelled by 106 triangular elements as shown in Figures 8a and 8b. The
characteristic length of the model elements was taken as 0.057 m, which gives
1 kHz as
a high frequency limit based on the ~/6-criteria (fh of the array system was
1.1 kHz).
All nodes except the center of the loudspeaker diaphragm were modelled as a
rigid
boundary. In order to obtain the directional pattern of each loudspeaker Zõ in
the array
system, the calculation was carried out one by one with the complete system.
For
example, when the directional pattern of the first loudspeaker Z1 was
calculated, only
the loudspeaker diaphragm center of the first loudspeaker Z1 was activated and
other
nodes were inactive. The calculation plane was selected as a circle in the
plane of the
active node of the activated loudspeaker Z,,.
Optimal filters were calculated by two methods. With both methods the aim was
to obtain an array with a constant noise sensitivity NS of 20 dB over a large
frequency
range. With the first method it was assumed that every loudspeaker unit Zõ
behaves as a
monopole and the scattering effect of the geometry was ignored. With the other
method
the directional pattern of each unit and the effect of scattering was taken
into account
both in the design of the optimized filters and in the computation of the
directivity
index DI and noise sensitivity NS.
From these designed filters the directivity index DI can be calculated in two
different ways. One way is to insert the filters and propagation factors
directly into
Equation (1). Another approach is to simulate a real measurement by inserting
the
required velocities at the loudspeaker diaphragm centers in the BEM model and
than to
compute the far field response in different directions. All four combinations
are
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presented in figure 9a. In addition, figure 9b shows the noise sensitivity NS
for the two
design methods, calculated with Equation (5).
Figures l0a and l Ob show the corresponding polar diagrams based on the same
methods as those of figure 9a: figure l0a shows the situation in which a
filter is applied
under simple source assumption and figure l Ob under considering the
directivity of the
loudspeakers Zõ . The predicted values from calculations with Equation (1)
show a
considerable positive influence due to the directivity of the loudspeakers Zõ
at lower
frequencies, but the directivity index DI is considerably lower when the BEM-
calculation method is applied. With the BEM method it is seen that the filters
that
include the directivity of the loudspeakers Zõ result in higher directivity
index DI
values at almost the whole frequency range compared to the case of the filters
derived
under simple source assumptions. This is probably due to the high mutual
screening of
the loudspeakers Zõ in this case.
Measurements
In order to observe the performance of the designed filters in a real
situation,
measurements were carried out under anechoic conditions. The size of the
loudspeakers Zõ and the geometry were the same as in figure 8. The filters of
the
constant beam width array that was introduced above was applied to this
system. The
filters were derived by two methods: the first design was based on the simple
source
assumption (monopole) and the second design was based on the loudspeaker
directivity
as obtained from the BEM simulation. The target value of the directivity index
DI was
chosen to be 12 dB.
Figures 11 a, l 1b, and l 1 c show measured directional patterns of the
prototype
endfire array with constant beam width. Figure 11 a shows a grey scale picture
of
directivity index in dB as a function of both frequency and direction for the
case of a
simple source assumption. Figure 1 lb shows the same as figure 1 lb but then
using
directivity of a single source obtained by a numerical model. Figure 1 l c
shows a
comparison of directivity index DI for different filters as a function of
frequency.
Taking into account the directivity of the loudspeakers Zõ (Figure 1 lb) shows
better results than when simple monopole behaviour of the loudspeakers Zõ is
assumed
(Figure 11a), however, it still has a higher sound level in off-axis
directions than
expected from the theoretical prediction in Figure 7. Figure 11 c shows a
comparison of
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directivity indexes DI's. Both measured cases show lower directivity index DI
values
than the target value of 12 dB, however the case using the filter considering
the
directivity of the loudspeakers Zõ has a higher and more stable directivity
index DI as
compared to the case using the filters derived under simple source
assumptions.
Conclusion
In the study performed by the inventor, the basic theory of an endfire
loudspeaker
array system is investigated and the effect of design parameters, number of
loudspeaker
units, their spacing, length of the array, and the use of the stability factor
of the optimal
beamformer are observed. The number of loudspeakers determines the maximum
value
of the directivity index DI, and the same directional characteristics are
observed
according to the frequency normalized by the high frequency limit. Increasing
of the
stability factor,Q causes a higher suppression of both the directivity index
DI and noise
sensitivity NS, however, this only applies below the frequency of maximum
directivity
index DI. To select the optimal value of the stability factor,Q for a given
target value,
the DI-NS plot is applied. Array length and number of loudspeakers are often
limited
by available budget and space. Therefore the stability factor,Q can be a
useful parameter
to control the directional characteristics of the array. As an example, a
constant beam
width array system is designed by the proper selection of stability factors.
Moreover,
the directional pattern considering the effect of other loudspeakers is
applied to the
optimal filter design to obtain an even better optimized filter. Preliminary
measurements on a prototype array system show that the directivity index DI's
are
lower than those of the simulations but they are promising for further
research on
optimization of this kind of endfire loudspeaker array systems.
REFERENCES
[1] I. Merks, Binaural application of microphone arrays for improved speech
intelligibility in a noisy environment, Ph.D. thesis, Technical University of
Delft
(2000).
[2] M. Brandstein and D. Ward, Microphone Arrays, Chap. 2 (Springer, New York,
2001).
[3] E. N. Gilbert and S. P. Morgan, "Optimum design of directive antenna
arrays
subject to random variations," Bell Syst. Tech. J., 34, 637-663 (1955).
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[4] M.M. Boone, "Directivity measurements on a highly directive hearing aid:
the
hearing glasses", 120'h AES Convention, Paris, 2006 May 20-23, paper nr. 6829.
[5] H. Cox, R. M. Zeskind and T. Kooij, "Practical supergain," IEEE Trans. on
Acoust.
Speech Signal Processing, 34, 393-398 (1986).
[6] L. E. Kinsler, A. R. Frey, A. B. Coppens and J. V. Sanders, Fundamentals
of
Acoustics, Chap .7 (John Wiley & Sons, New York, 2000).
[7] E. G. Williams, Fourier Acoustics - Sound Radiation and Nearfield
Acoustical
Holography, Chap. 6 (Academic Press, London, 1999).
